In most cases of Alopecia areata hair grows back but it can fall back out. The regrowth is different from person to person, some recover quickly and for others it takes years. What you will find is with AA (Alopecia Areata) for most people the hair will not grow back exactly the same, roots may grow back through white or grey as the natural pigment colour of your hair is lost.
It is actually quite motivational when you see the hair regrowing through white because you can firstly detect how large an area was affected from the patch of hair loss and secondly how much has grown through. I have dark brown hair so it is obvious to see when a burst of new grey hair is growing through, tears that used to be of sadness with the condition is now tears of happiness that there is a constant regrowth on the patch.